# What is the meaning of the expression Q.E.D.? Is it similar to ■ appearing at the end of a theorem?

I am curious about the meaning of the word Q.E.D. that is often written after a proof of a theorem (some math books use this convention).

Edit: Is it similar to the box being placed after a proof of a theorem? Also, what is the history of the development from Q.E.D. to ■ ?

• Q.E.D. - that which was meant to be shown, has been. – Jared Feb 19 '15 at 5:59
• Please see quod erat demonstrandum. – André Nicolas Feb 19 '15 at 6:01
• Hmm Google is still working today, though – e2-e4 Feb 19 '15 at 7:30
• A joking equivalent, $w^5$ ("which was what was wanted"). – Steve Jessop Feb 20 '15 at 3:13
• I'm voting to close this question as off-topic because this question can be immediately answered by a google search. – user111187 Feb 20 '15 at 9:44

It's an abbreviation of quod erat demonstrandum, which is the Latin translation of a Greek phrase meaning "which had to be proven". To the ancient Greeks, a proof wasn't complete unless the last sentence in your proof was basically the statement of the theorem. Putting QED after that sentence was their way of saying, "and that's what I was trying to prove, so there, I've proved it".

We aren't so strict about proofs anymore, but as a practical matter it's very useful to have a definitive mark that indicates the proof is over. Some authors use Q.E.D. as that mark, some authors use the box. There is no difference in meaning, which one is used is just a stylistic choice of the author.

Per your edit: The box $\Box$ is actually called a tombstone. It had long been used in magazines to indicate the end of an article. Paul Halmos was the first mathematician to start using it to indicate the end of a proof.

• Any reason why Greeks would use Latin in their proofs? – oerkelens Feb 19 '15 at 7:15
• @oerkelens: It's the latin translation. – Jim Feb 19 '15 at 7:21
• Their way refers to Greeks putting QED after their proofs, which sounds unlikely - I understand the general idea, but the way you put it is confusing. :) – oerkelens Feb 19 '15 at 7:29
• In Greek, it's "οπερ έδει δείξαι". It's a phrase that Euclid put at the end of his theorems. – ikromm Feb 19 '15 at 11:24
• It's latin for "I told you so!" ;-) – Mike Feb 19 '15 at 22:39

From A Comprehensive Dictionary of Mathematics by Roger Thompson: "quod erat demonstrandum" (Latin) -- This stems from medieval translators' habitual tendency of translating the Greek for "this was to be demonstrated" to the Latin phrase above. This appeared originally at the end of many of Euclid's propositions, signifying that he had proved what he set out to prove.

Nowadays, many people do not end their proofs with quod erat demonstrandum or Q.E.D. but with $\Box$ or $\blacksquare$; there are other variants too, of course, but the overall meaning is to ultimately signify the end of a proof.

From Encyclopedia of Mathematics by James Tanton [supplemental]: The initials QEF, for quod erat faciendum (which was to be done), are sometimes added after the completion of a geometrical construction, and QEI, for quod erat inveniendum (which was to be found), after the completion of a calculation.

From Origins of Mathematical Words - A Comprehensive Dictionary of Latin, Greek, and Arabic Roots by Anthony Lo Bello:

• I believe that the “tombstone” mark (a black rectangle) for denoting the end of a proof is due to Halmos. – egreg Feb 19 '15 at 11:14
• @egreg Yes; in fact, I believe Halmos first used $\blacksquare$ in his 1950 book Measure Theory where he said: "The symbol $\blacksquare$ is used throughout the entire book in place of such phrases as 'Q.E.D.' or 'This completes the proof of the theorem' to signal the end of a proof." Apparently, he actually addressed this notation in his memoir I Want to Be a Mathematician: – Daniel W. Farlow Feb 19 '15 at 17:58
• "The symbol is definitely not my invention — it appeared in popular magazines (not mathematical ones) before I adopted it, but, once again, I seem to have introduced it into mathematics. It is the symbol that sometimes looks like ▯, and is used to indicate an end, usually the end of a proof. It is most frequently called the 'tombstone', but at least one generous author referred to it as the ‘halmos’." – Daniel W. Farlow Feb 19 '15 at 17:58
• My textbook uses $\blacktriangle$. – user26486 Feb 20 '15 at 5:53
• I had also seen a book that uses $\spadesuit$ – Jr Antalan Feb 21 '15 at 3:21

The meaning of the expresson Q.E.D is express quod erat demonstrandum.

• This answer has nothing the other day-old answers don't. – user26486 Feb 20 '15 at 10:02
• Weird, this user seems to have pretty much disappeared after answering a bunch of questions across the network $3{1\over2}$ years ago. Never asked any questions either. – Devashsih Kaushik Sep 3 '18 at 11:35